The inverse image of dense set is dense and of a comeager set is comeager?.

Let $X,Y$ be topological and $f:X \to Y$ be open and continuos.

I am studying Baire space and I would like to try the following facts :

$(i)$ The inverse image of dense set is dense.

$(ii)$ The inverse image of comeager set is comeager.

I am studying Baire space and I would like to try the following facts

Can we collaborate to verify the above claims?

Thanks.

Solution 1:

(i) Yes. Let $D’$ be a dense subset of the space $Y$ and $U$ be a non-empty open subset of a space $X$. Then $f(U)$ is a non-empty open subset of a space $Y$, so it contains a point $d’\in D$. Since $d’\in f(U)$, there is a point $d\in U$ such $f(d)=d’$. So $d\in f^{-1}(D)\cap U\ne\varnothing$.

(ii) Yes. Let $A’$ be a comeager subset of the space $Y$. Then $Y\setminus A’$ is a countable union $\bigcup B’_n$ of nowhere dense subsets of the space $Y$. For each $n$ put $B_n=f^{-1}(B’_n)$. If some $B_n$ is dense in a non-empty open subset $U$ of a space $X$, then $B’_n=f(B_n)$ is dense in a non-empty open subset $f(U)$ of a space $X$, a contradiction. So each $B_n$ is a nowhere dense subset of the space $X$ and the union $B=\bigcup B_n $ is a meager subset of the space $X$. Thus $X\setminus B=f^{-1}(A’)$ is a comeager subset of the space $X$.